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Bias and Measurement Errors
Astro193, March 4 2015


Measurement Error Effects
· Blurring and broadening distribution of quantities
· Measured is broadly scattered in comparison to True

·

Impacts statistical inference: · Biased measurements · Smeared out any trends in the data
­ the separation in the True is not present in the Measured

Brandon Kelly 2011, Measurement Error Models in Astronomy SCAMA V, (astro-ph 1112.1745)

Covariate - independent variable, e.g. mass, color, energy Response - dependent variable, e.g. luminosity, color


Impact on Linear Regression
Assuming the homoskedastic error and using least-squares the results on uncertainty of the slope () assuming a fixed intercept ():

Flatter trend Larger intrinsic dispersion
Uncertainty on the slope goes to 0 when x large


Unbiased Approach - I
· Method of Moments (MoM)
· Least-square estimate of slope derived from moments of distribution · Corrects moments of measured data for measurement error Ordinary Least Squares

Method of Moments


Simulation Test

BCES - Akritas & Bershady (ApJ 1996)

Kelly 2011


Unbiased approach - II Bayesian Methods
Posterior Median FITEXY Posterior Median

95%

Errorbars omiited for clarity

Example: Studying trends in X-ray photon index observed for a sample
of radio quiet quasars with their Eddington Luminosity.
Note: - photon index is based on X-ray spectrum with the uncertainty from fitting the spectum, - the Eddington luminosity LEdd error is dominated by the black hole mass measurement error

Kelly, 2007, ApJ. 665,1489